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Overhead-Aware Design of Reconfigurable Intelligent Surfaces in Smart Radio Environments

Alessio Zappone, Marco Di Renzo, Farshad Shams, Xuewen Qian, Merouane Debbah

arXiv:2003.02538v3cs.IT

TL;DR

The paper addresses the missing treatment of channel-estimation and RIS-configuration overhead in RIS resource allocation. It develops an overhead-aware framework that jointly optimizes RIS phases, transceiver filters, powers, and bandwidths, including rate-energy trade-offs. Results show that RIS optimization is beneficial under low antenna counts or suitable feedback, but its overhead can reduce or reverse the gain as antenna counts and estimation costs increase.

  • Problem

    Previous RIS resource-allocation studies optimize communication resources without explicitly accounting for channel-estimation and optimized phase-configuration overhead.

  • Method

    The paper incorporates channel-estimation and RIS-configuration overhead into rate and energy-efficiency expressions, then optimizes phases, transmit and receive filters, powers, bandwidths, and their rate-energy trade-off.

  • Results

    RIS optimization improves performance with suitable feedback mechanisms or few antennas, whereas higher antenna counts, longer estimation times, or feedback costs can make non-optimized RIS operation preferable.

  • Takeaways & Limitations

    RIS deployment requires balancing radio-resource optimization against the overhead of estimating channels and communicating optimized phase configurations.

Abstract

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Reconfigurable intelligent surfaces have emerged as a promising technology for future wireless networks. Given that a large number of reflecting elements is typically used, and that the surface has no signal processing capabilities, a major challenge is to cope with the overhead that is required to estimate the channel state information and to report the optimized phase shifts to the surface. This issue has not been addressed by previous works, which do not explicitly consider the overhead during the resource allocation phase. This work aims at filling this gap, developing an overhead-aware resource allocation framework for wireless networks where reconfigurable intelligent surfaces are used to improve the communication performance. An overhead model is developed and incorporated in the expressions of the system rate and energy efficiencies, which are then optimized with respect to the phase shifts of the reconfigurable intelligent surface, the transmit and receive filters, and the power and bandwidth used for the communication and feedback phases. The bi-objective maximization of the rate and energy efficiency is carried out as well. The proposed framework allows characterizing the trade-off between optimized radio resources and the related overhead in networks with reconfigurable intelligent surfaces.

I. INTRODUCTION

The paper addresses RIS resource allocation while explicitly accounting for channel-estimation and phase-configuration overhead, a gap in prior optimization studies. It develops closed-form and globally optimized resource-allocation methods for RIS phases, filters, powers, and bandwidths, and evaluates their performance and rate-energy trade-offs.

  • Motivation and gap: Prior RIS resource-allocation studies optimize the communication phase without accounting for channel-estimation and phase-configuration overhead.The paper identifies this omission as especially important because RISs may contain many distributed passive elements.
  • Framework: The proposed framework models overhead for channel estimation and RIS configuration and incorporates it into rate and energy-efficiency optimization.The framework allocates resources across communication and feedback phases rather than treating feedback as cost-free.
  • Optimization methods: Two closed-form methods jointly optimize RIS phase shifts, transmit precoding, and receive decoding filters, with provable optimality for rank-one channels.Rank-one channels include the single-antenna transmitter and receiver case.
  • Optimization methods: Power and bandwidth allocations for rate, energy efficiency, and their trade-off are obtained through globally optimal convex or pseudo-convex optimization methods.These algorithms are designed for the communication and feedback resources and have limited complexity.
  • Evaluation: The numerical evaluation compares the proposed closed-form phase optimization with alternating optimization and reports similar performance for the two approaches.The paper also presents numerical results for spectral efficiency, energy efficiency, and their rate-energy trade-off.
  • System model: The system model uses a point-to-point single-stream RIS link with multiple transmit and receive antennas, no direct transmitter-receiver link, and individually phase-controlled passive scatterers.The RIS applies independently selected phase shifts through a diagonal configuration matrix, while channel estimation and configuration occur outside the RIS.

III. OPTIMIZATION OF Φ, q, w

The section optimizes the RIS phase matrix and transmit/receive filters by bounding the objective, using singular-vector choices and closed-form phase alignment. The resulting upper-bound maximizer is characterized analytically.

  • Objective formulation: Rate and energy-efficiency maximization over Φ, q, and w reduces to optimizing the numerator because these variables do not appear in the energy-efficiency denominator.The numerator coincides with the system rate for fixed p, pF, B, and BF.
  • Upper-bound optimization: For fixed Φ, the optimal q and w are the dominant right and left singular vectors of GΦH.This follows from the spectral-norm characterization of the maximum over unit-norm transmit and receive vectors.
  • Upper-bound optimization: Maximizing the largest singular value of GΦH directly over Φ is considered prohibitive and does not yield a closed-form solution.The paper therefore develops closed-form approaches for optimizing an upper or lower bound.
  • Upper-bound optimization: Cauchy-Schwarz-based inequalities and a simplex bound establish the upper-bound construction used by the closed-form solution.The proof invokes the triangle inequality, Cauchy-Schwarz inequality, and Lemma 1.
  • Upper-bound optimization: The upper-bound maximizer sets q and w to selected singular vectors and chooses each RIS phase shift to compensate component phase differences.The proposition gives the corresponding closed-form choices for q, w, and φn.

B. Optimizing a lower-bound of the objective of (4)

The lower-bound approach fixes the beamforming and decoding vectors, aligns RIS phases elementwise, and then selects beamformers from the dominant singular directions of an aggregated matrix.

  • Lower-bound construction: For fixed q and w, each RIS phase is chosen to align the corresponding cascaded channel contributions.The phase rule uses the complex quantities formed from GHw and Hq.
  • Lower-bound construction: The final maximization selects q and w as the dominant right and left eigenvectors of the aggregated matrix formed from the aligned RIS contributions.The resulting phase choice is retained through the elementwise alignment rule.

C. Tackling (4) by alternating maximization

The paper also solves the original objective by alternating between RIS phase updates and singular-vector updates, while modeling channel-estimation and RIS-configuration overhead separately.

  • Alternating maximization: With Φ fixed, the optimal w and q are the dominant left and right eigenvectors of A = GΦH.With w and q fixed, the RIS phases are updated using the elementwise phase-alignment rule.
  • Alternating maximization: Algorithm 1 alternates RIS phase updates with eigenvector updates until convergence.It initializes feasible w and q, updates all φn, forms A, and resets w and q to A’s dominant eigenvectors.
  • Overhead modeling: Channel estimation and RIS phase configuration occur before data transmission and cannot be performed at the RIS itself.They may instead be performed at the transmitter or receiver, while the RIS has minimal processing capabilities.
  • Overhead modeling: The feedback duration depends on feedback power and bandwidth, complicating joint optimization of communication and feedback resources.The paper addresses optimization of p, pF, B, and BF for rate, energy efficiency, and their trade-off.
  • Overhead modeling: The overhead model neglects feedback of the receive filter w because it is typically negligible relative to RIS phase-shift feedback.The stated focus is the feedback required to operate the RIS.
  • Overhead modeling: Channel-estimation overhead depends on the protocol: sequential pilots require TE = (NTNR + 1)T0, whereas parallel pilots give TE = (N + 1)T0.The corresponding parallel-pilot estimation power is also specified in the model.

A. Rate maximization

The rate-maximization problem is difficult in its original product form, but concavity results permit an equivalent convex reformulation and a reduced two-variable optimization.

  • Problem structure: The original rate objective is not jointly concave because it is a product of two functions involving communication and feedback resources.This prevents direct use of standard convex optimization algorithms.
  • Concavity analysis: R is jointly increasing and jointly concave in (p, B), and separately jointly increasing and jointly concave in (pF, BF).These properties are established in Lemmas 2 and 3.
  • Convex reformulation: Taking the logarithm yields an equivalent convex optimization problem with linear resource constraints and a convex feedback-related constraint.The reformulation has the same solutions as the original rate problem.
  • Variable reduction: Because the objective increases in all arguments while the feedback constraint decreases in pF and BF, the power and bandwidth constraints are tight at optimum.The problem can therefore substitute pF = Pmax − p and BF = Bmax − B.
  • Variable reduction: The reduced problem has only two optimization variables, p and B, with feedback resources recovered by subtraction from the total budgets.Linear variable transformations preserve convexity.
  • Subproblem solutions: The fixed-bandwidth subproblem has the unique solution p* = min(p̄, Pmax − pmin), where p̄ is the unique stationary point.Strict concavity establishes uniqueness.
  • Subproblem solutions: The fixed-power subproblem likewise has the unique solution B* = min(B̄, Bmax − pB), where B̄ is its unique stationary point.Its objective is strictly concave and changes from increasing to decreasing at B̄.

B. Energy efficiency optimization

Energy-efficiency optimization is reformulated into a convex-feasibility problem and solved through line search over an auxiliary variable with fractional programming. The relaxed formulation is equivalent to the original problem and enables Algorithm 2.

  • Energy efficiency increases with B and B_F, but is not generally monotonic in p or p_F, so full power use is not guaranteed at the optimum.
  • Problem (28) and Problem (29) have the same set of optimal solutions.
  • Problem (29) remains challenging because its objective lacks the concavity and convexity properties required for direct fractional programming.For fixed y, it becomes a pseudo-concave maximization problem with a concave numerator and affine denominator.
  • For fixed y, the energy-efficiency subproblem can be solved with a fractional-programming method such as Dinkelbach’s method.
  • The solution is obtained by line-searching over y and solving the corresponding pseudo-concave maximization problem at each trial value.

C. Rate-EE optimization

The rate–energy-efficiency problem characterizes Pareto-optimal resource allocations between the two objectives. It uses an epigraph reformulation, line search over an auxiliary variable, and bisection over the objective variable.

  • The bi-objective problem characterizes the Pareto-optimal frontier between system rate and energy efficiency.
  • Rate and energy efficiency generally require different resource allocations, creating a nontrivial trade-off between the objectives.
  • Weighted scalarization with α∈(0,1) yields Pareto-optimal solutions, and sweeping α produces the Pareto frontier.
  • The epigraph formulation maximizes t subject to power, bandwidth, and auxiliary-variable constraints.
  • For fixed y, the reformulated problem is solved by bisection over t inside a line search over y.

V. OPTIMALITY PROPERTIES AND COMPUTATIONAL

This section introduces the rate–energy-efficiency maximization algorithm and describes its iterative parameter sweep and bisection-based subproblem solution.

  • Algorithm 3 performs M iterations over the scalarization parameter and solves Problem (36) by bisection over t.
  • The algorithm selects the iteration with the largest objective value and outputs its associated allocation.

A. Algorithms for the optimization of Φ, q, w

The proposed RIS, transmit-beamforming, and receive-filter methods use closed-form, non-iterative optimization rather than alternating optimization. They reduce computational complexity, but are globally optimal only under rank-one channel conditions.

  • The proposed algorithms are globally optimal when both channel matrices have rank one; otherwise, their bound-based solutions are generally not globally optimal.The rank-one case includes single-antenna transmission and many mmWave channels.
  • Joint global optimization of Φ, q, and w is computationally prohibitive because it requires exhaustive search without a tractable dominant-singular-value expression.
  • Alternating optimization repeatedly computes an SVD and RIS phase adjustments, whereas the proposed methods require one SVD and one phase adjustment.
  • The proposed non-iterative methods reduce complexity relative to alternating optimization by a factor of N_it and perform close to it numerically.

B. Algorithms for the optimization of p, pF , B, BF

The algorithms globally optimize transmit and feedback powers and bandwidths for rate, energy efficiency, and their joint trade-off, using tractable reformulations and search procedures.

  • Rate optimization is recast as a concave maximization solvable with polynomial complexity in the number of optimization variables.
  • Energy-efficiency maximization uses a scalar line search over auxiliary variable y and pseudo-concave maximization at each tested value.Its overall complexity is polynomial in the optimization variables and linear in the line-search range.
  • Bi-objective rate–energy-efficiency maximization becomes a sequence of feasibility tests that are convex when y is fixed.The procedure is polynomial in the optimization variables and linear in the number M of line-search points, with the optimal parameter t found by bisection.

VI. NUMERICAL RESULTS

Numerical results show that RIS optimization can improve spectral and energy efficiency when overhead is modest, but increasing antennas or estimation time can make feedback overhead outweigh the gains.

  • RIS optimization outperforms no RIS optimization for single-antenna links even after channel-estimation and configuration overhead is included.Closed-form Schemes (b) and (c) perform similarly to alternating optimization and are provably optimal when NT = NR = 1.
  • For NT = 1 and NR = 8, the performance gap between optimized and non-optimized RIS schemes becomes smaller because feedback data increases.Avoiding phase optimization avoids estimating H and G for each individual phase shift.
  • When NT = NR = 8 and T0 = 0.8 µs, no RIS optimization outperforms RIS-based schemes because longer channel estimation increases overhead.With T0 = 0.15 µs, resource allocation remains beneficial up to N = 130, after which no RIS becomes better; Schemes (b) and (d) perform similarly.
  • The results motivate RIS deployment with few transmit and receive antennas, especially for large N, because each additional antenna requires N new channel estimates and phase reports.The optimized comparisons indicate that a moderate number of antennas does not necessarily improve performance, and RISs may reduce the need for beamforming and combining.
  • For energy-efficiency optimization, no RIS feedback can outperform optimized schemes when NR = 8, NT = 1, T0 = 0.8 µs, and N > 150.Feedback overhead reduces both the rate numerator and the power-consumption denominator of energy efficiency; the study also evaluates rate–energy Pareto boundaries.
  • With simultaneous orthogonal pilots, optimized RIS provides higher spectral efficiency in the tested NR = 8, NT = 1 and NT = NR = 8 cases.Under the same challenging overhead setting, energy-efficiency optimization becomes unfavorable for NT = NR = 8 and N ≥ 150, while RIS optimization improves the spectral-energy trade-off for N = 20 and N = 100.

VII. CONCLUSIONS

The paper develops an overhead-aware RIS resource-allocation framework covering spectral efficiency, energy efficiency, and their trade-off. Its conclusions emphasize tractable optimization while identifying multi-user interference as a complication.

  • The framework optimizes RIS phase shifts, transmit and receive vectors, communication and feedback powers, and bandwidths for spectral and energy efficiency.Phase and filtering methods are closed-form, while power and bandwidth allocation uses concave or pseudo-concave maximization.
  • Multi-user interference complicates resource allocation and may require numerical optimization techniques.
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